Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10144 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,3)(4,7,5,8)(11,17)(12,19)(14,22)(15,20)(18,21)(23,24);(2,7)(3,8)(4,5)(9,10,13,16)(11,14,18,23)(12,20,19,15)(17,24,21,22);(1,3,6,2)(4,8,5,7)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,19)(20,22)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,c^b=cf,d^b=dg,e^b=egh,f^b=fg,d^c=dh,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10159, #236913, #236926, … |
| maximal normal subgroup of | C2×#10159, #236913, #236926, … |
| maximal quotient of | #102598, #102611, #102617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3147);